To prepare yogurt cheese, place a colander in a 2-quart glass measure or medium bowl. Line colander with 4 layers of cheesecloth, allowing cheesecloth to extend over outside edges. Spoon yogurt into colander. Cover loosely with plastic wrap, and refrigerate 12 hours. Spoon yogurt cheese into a bowl; discard liquid. Cover and refrigerate.
To prepare ice cream, combine 1/4 cup sugar and eggs in a medium bowl, stirring well with a whisk.
Combine milk and half-and-half in a heavy saucepan over medium-high heat. Scrape seeds from vanilla bean, and add seeds and bean to milk mixture.
Heat to 180 or until tiny bubbles form around edge of pan, stirring frequently (do not boil). Gradually add milk mixture to egg mixture, stirring constantly with a whisk. Return milk mixture to pan; cook over medium heat until thick (about 5 minutes), stirring constantly.
Remove from heat.
Place pan in a large ice-filled bowl until mixture cools to room temperature (about 25 minutes), stirring occasionally.
Remove vanilla bean.
Pour mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon ice cream into a freezer-safe container; cover and freeze 1 hour or until firm.
To prepare the sauce, place 1/3 cup sugar in a small, heavy saucepan over medium-high heat; cook until sugar dissolves, stirring as needed to dissolve sugar evenly (about 6 minutes). Continue cooking about 1 minute or until golden.
Remove from heat; carefully stir in water (caramelized sugar will seize slightly and stick to spoon).
Place pan over medium-high heat until caramelized sugar melts. Scrape seeds from vanilla bean; add seeds and bean to mixture. Cook 2 minutes or until reduced to 6 tablespoons, stirring occasionally.
Remove from heat, and let stand 10 minutes. Discard vanilla bean.
Place 1/4 cup yogurt cheese into each of 6 bowls; top with 1 tablespoon sauce and 1/2 cup ice cream.
